Updated on 2024/02/28

写真a

 
EDA, Katsuya
 
Affiliation
Faculty of Science and Engineering
Job title
Professor Emeritus
Degree
理学博士 ( 筑波大学 )

Education Background

  •  
    -
    1971

    Tokyo University of Education   Graduate School, Division of Natural Science  

Professional Memberships

  •  
     
     

    日本数学会

Research Areas

  • Geometry / Applied mathematics and statistics / Basic mathematics

Research Interests

  • Applications of Set Theory

 

Papers

  • Existence and uniqueness of group structures on covering spaces over groups

    Katsuya Eda, Vlasta Matijevic

    FUNDAMENTA MATHEMATICAE   238 ( 3 ) 241 - 267  2017  [Refereed]

     View Summary

    Let f : X -> Y be a covering map from a connected space X onto a topological group Y and let x(0) is an element of X be a point such that f(x(0)) is the identity of Y. We examine if there exists a group operation on X which makes X a topological group with identity x(0) and f a homomorphism of groups. We prove that the answer is positive in two cases: if f is an overlay map over a locally compact group Y, and if Y is locally compactly connected. In this way we generalize previous results for overlay maps over compact groups and covering maps over locally path-connected groups. Furthermore, we prove that in both cases the group structure on X is unique.

    DOI

  • Cotorsion-free groups from a topological viewpoint

    Katsuya Eda, Hanspeter Fischer

    TOPOLOGY AND ITS APPLICATIONS   214   21 - 34  2016.12  [Refereed]

     View Summary

    We present a characterization of cotorsion-free abelian groups in terms of homomorphisms from fundamental groups of Peano continua, which aligns naturally with the generalization of slenderness to non-abelian groups. In the process, we calculate the first homology group of the Griffiths twin cone. (C) 2016 Elsevier B.V. All rights reserved.

    DOI

  • Singular homology groups of one-dimensional Peano continua

    K. Eda

    FUNDAMENTA MATHEMATICAE   232 ( 2 ) 99 - 115  2016  [Refereed]

     View Summary

    Let X be a one-dimensional Peano continuum. Then the singular homology group H-1(X) is isomorphic to a free abelian group of finite rank or the singular homology group of the Hawaiian earring.

    DOI

  • Maps from the minimal grope to an arbitrary grope

    Matija Cencelj, Katsuya Eda, Aleš Vavpetič

    International Journal of Algebra and Computation   23 ( 3 ) 503 - 519  2013.05  [Refereed]

     View Summary

    We consider open infinite gropes and prove that every continuous map from the minimal grope to another grope is nulhomotopic unless the other grope has a «branch» which is a copy of the minimal grope. Since every grope is the classifying space of its fundamental group, the problem is translated to group theory and a suitable block cancellation of words is used to obtain the result. © 2013 World Scientific Publishing Company.

    DOI

  • On 2-dimensional Nonaspherical Cell-like Peano Continua: A Simplified Approach

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    MEDITERRANEAN JOURNAL OF MATHEMATICS   10 ( 1 ) 519 - 528  2013.02  [Refereed]

     View Summary

    We construct a functor AC(-, -) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)-dimensional; and (iii) For a path connected space X, pi (1)(X, x) is trivial if and only if pi (2)(AC(X, x)) is trivial. As a corollary, AC(S (1), x) is a 2-dimensional nonaspherical cell-like Peano continuum.

    DOI

  • Covering maps over solenoids which are not covering homomorphisms

    Katsuya Eda, Vlasta Matijevic

    FUNDAMENTA MATHEMATICAE   221 ( 1 ) 69 - 82  2013  [Refereed]

     View Summary

    Let Y be a connected group and let f : X -> Y be a covering map with the total space X being connected: We consider the following question: Is it possible to define a topological group structure on X in such a way that f becomes a homomorphism of topological groups. This holds in some particular cases: if Y is a pathwise connected and locally pathwise connected group or if f is a finite-sheeted covering map over a compact connected group Y. However, using shape-theoretic techniques and Fox's notion of an overlay map, we answer the question in the negative. We consider infinite-sheeted covering maps over solenoids, i.e. compact connected 1-dimensional abelian groups. First we show that an infinite-sheeted covering map f : X -> Sigma with a total space being connected over a solenoid Sigma does not admit a topological group structure on X such that f becomes a homomorphism. Then, for an arbitrary solenoid Sigma, we construct a connected space X and an infinite-sheeted covering map f : X -> Sigma, which provides a negative answer to the question.

    DOI

  • On snake cones, alternating cones and related constructions

    Katsuya Eda, Umed H. Karimov, Dušan Repovš, Andreas Zastrow

    Glasnik Matematicki   48 ( 1 ) 115 - 135  2013  [Refereed]

     View Summary

    We show that the Snake on a square SC(S1) is homotopy equivalent to the space AC(S1) which was investigated in the previous work by Eda, Karimov and Repovš. We also introduce related constructions CSC(-) and CAC(-) and investigate homotopical differences between these four constructions. Finally, we explicitly describe the second homology group of the Hawaiian tori wedge.

    DOI

  • The classification of the inverse limits of sequences of free groups of finite rank

    Katsuya Eda, Jun Nakamura

    Bulletin of the London Mathematical Society   45 ( 4 ) 671 - 676  2013  [Refereed]

     View Summary

    We classify the groups that are the inverse limits of sequences of free groups of finite rank. Except free groups of finite rank, there exist four groups that are the inverse limits of sequences of free groups of finite rank. © 2013 London Mathematical Society.

    DOI

  • Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces

    Katsuya Eda

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   63 ( 3 ) 769 - 787  2011.07  [Refereed]

     View Summary

    We strengthen previous results on the fundamental groups of the Hawaiian earring and wild Peano continua. Let X be a path-connected, locally path-connected, first countable space which is not locally semi-simply connected at any point. If the fundamental group pi(1)(X) is a subgroup of a free product *H-j is an element of J(j), then it is contained in a conjugate subgroup to some H-j.

    DOI

  • On the Singular Homology of One Class of Simply-connected Cell-like Spaces

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    MEDITERRANEAN JOURNAL OF MATHEMATICS   8 ( 2 ) 153 - 160  2011.06  [Refereed]

     View Summary

    In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called Snake space. In the sequel we introduced the functor SC(-,-) defined on the category of all spaces with base points and continuous mappings. For the circle S (1), the space SC (S-1, *) is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces SC (Z, *) for any path-connected compact spaces Z.

    DOI

  • Atomic property of the fundamental groups of the Hawaiian earring and wild Peano continua

    EDA, Katsuya

    J. Math. Soc. Japan   ( 63 ) 769 - 787  2011  [Refereed]

    DOI

  • ON THE ASPHERICITY OF ONE-POINT UNIONS OF CONES

    Katsuya Eda, Kazuhiro Kawamura

    TOPOLOGY PROCEEDINGS, VOL 36   36   63 - +  2010

     View Summary

    We prove that the one-point union of two copies of the cone over the Hawaiian earring is aspherical.

  • Homotopy types of one-dimensional Peano continua

    Katsuya Eda

    FUNDAMENTA MATHEMATICAE   209 ( 1 ) 27 - 42  2010

     View Summary

    Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.

    DOI

  • ON THE SECOND HOMOTOPY GROUP OF SC(Z)

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    GLASNIK MATEMATICKI   44 ( 2 ) 493 - 498  2009  [Refereed]

     View Summary

    In our earlier paper we introduced a cone-like space SC(Z). In the present note we establish some new algebraic properties of SC(Z).

  • A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    TOPOLOGY AND ITS APPLICATIONS   156 ( 3 ) 515 - 521  2009.01

     View Summary

    We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950s. (C) 2008 Elsevier B.V. All rights reserved.

    DOI

  • Finite index supergroups and subgroups of torsion free abelian groups of rank two

    Katsuya Eda, Vlasta Matijevic

    JOURNAL OF ALGEBRA   319 ( 9 ) 3567 - 3587  2008.05

     View Summary

    Every torsionfree abelian group A of rank two is a subgroup of Q circle plus Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2 x 2-matrices as the bonding maps. Using these direct systems we classify all subgroups of Q circle plus Q which are finite index supergroups of A or finite index subgroups of A. Using this classification we prove that for each prime p there exists a torsionfree abelian group A satisfying the following, where A <= Q circle plus Q and all supergroups are subgroups of Q circle plus Q:
    (1) for each natural numbers there are Sigma(q vertical bar s),(gcd(p,q)=1) q s-index supergroups and also Sigma(q vertical bar s),(gcd(p,q)=1) s-index subgroups;
    (2) each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index, subgroups are non-isonnorphic. (c) 2008 Elsevier Inc. All rights reserved.

    DOI

  • On the fundamental group of ℝ3 modulo the case-Chamberlin continuum

    Katsuya Eda, Umed H. Karimov, Dušan Repovš

    Glasnik Matematicki   42 ( 1 ) 89 - 94  2007.06

     View Summary

    It has been known for a long time that the fundamental group of the quotient of ℝ3 by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.

    DOI

  • A construction of noncontractible simply connected cell-like two-dimensional Peano continua

    Katsuya Eda, Urned H. Karimov, Dusan Repovs

    FUNDAMENTA MATHEMATICAE   195 ( 3 ) 193 - 203  2007

     View Summary

    Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts from a noncontractible n-dimensional Peano continuum for any n > 0, then our construction yields a simply connected noncontractible (n + l)-dimensional cell-like Peano continuum. In particular, starting from the circle S-1, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum.

    DOI

  • Finite sheeted covering maps over 2-dimensional connected, compact Abelian groups

    K Eda, Matijevic, V

    TOPOLOGY AND ITS APPLICATIONS   153 ( 7 ) 1033 - 1045  2006.01

     View Summary

    Let G be a 2-dimensional connected, compact Abelian group and s be a positive integer. We prove that a classification of s-sheeted covering maps over G is reduced to a classification of s-index torsionfree supergroups of the Pontrjagin dual (G) over cap. Using group theoretic results from earlier paper we demonstrate its consequences. We also prove that for a connected compact group Y:
    (1) Every finite-sheeted covering map from a connected space over Y is equivalent to a covering homomorphism from a compact, connected group.
    (2) If two finite-sheeted covering homomorphisms over Y are equivalent, then they are equivalent as topological homomorphisms. (C) 2005 Elsevier B.V. All rights reserved.

    DOI

  • Torus-like continua which are not self-covering spaces

    K Eda, J Mandic, Matijevic, V

    TOPOLOGY AND ITS APPLICATIONS   153 ( 2-3 ) 359 - 369  2005.09

     View Summary

    For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T-2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f(0): X-0 -> Y, f(1) : X-1 -> Y, f(2): X-2 -> Y such that the total spaces X-0 = Y, X-1 and X-2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-shected covering map f(3): X-3 -> Y such that X-3 and Y are nonhomeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps. (c) 2004 Elsevier B.V. All rights reserved.

    DOI

  • Torus-like continua which are not self-covering spaces

    K Eda, J Mandic, Matijevic, V

    TOPOLOGY AND ITS APPLICATIONS   153 ( 2-3 ) 359 - 369  2005.09

     View Summary

    For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T-2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f(0): X-0 -> Y, f(1) : X-1 -> Y, f(2): X-2 -> Y such that the total spaces X-0 = Y, X-1 and X-2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-shected covering map f(3): X-3 -> Y such that X-3 and Y are nonhomeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps. (c) 2004 Elsevier B.V. All rights reserved.

    DOI

  • Algebraic topology of Peano continua

    K Eda

    TOPOLOGY AND ITS APPLICATIONS   153 ( 2-3 ) 213 - 226  2005.09

     View Summary

    Let X be a Peano continuum. Then the following hold: (1) The singular cohomology group H I (X) is isomorphic to the tech cohomology group h I (X). (2) For each homomorphism h : pi(1)(X) -> *(i is an element of l)G(i) there exists a finite subset F of I such that Im(h) subset of *(i is an element of F)G(i). (3) For each injective homomorphism h : pi(1) (X) -> G(0) * G(1) there exists a finitely generated subgroup F-0 of G(0) or a finitely generated subgroup F-1 of G(1) such that Im(h) subset of F-0 * G(1) or Im(h) subset of G(0) * F-1. (c) 2004 Published by Elsevier B.V.

    DOI

  • Fundamental groups having the whole information of spaces

    G Conner, K Eda

    TOPOLOGY AND ITS APPLICATIONS   146   317 - 328  2005.01

     View Summary

    We introduce a new construction of spaces from groups using homomorphic images of the fundamental group of the Hawaiian earring H. According to this construction the Menger sponge, Sierpinski gasket, Sierpinski carpet and the direct product of their countable many copies are recovered from their fundamental groups. (C) 2004 Published by Elsevier B.V

    DOI

  • The fundamental groups of one-dimensional spaces and spatial homomorphisms

    Katsuya Eda

    Topology and its Applications   123 ( 3 ) 479 - 505  2002.09

     View Summary

    Let X be a one-dimensional metric space and ℍ be the Hawaiian earring.(1) Each homomorphism from π1(ℍ) to π1(X) is induced from a continuous map up to the base-point-change isomorphism on π1(X).(2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of ℍ if and only if π1(X) is isomorphic to π1(ℍ), if and only if X has a unique point at which X is not semi-locally simply connected. (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if π1(X) and π1(Y) are isomorphic. Moreover, each isomorphism from π1(X) to π1(Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism. © 2001 Elsevier Science B.V. All rights reserved.

    DOI

  • Fundamental groups of one-dimensional spaces and spatial homomorphisms

    K Eda

    TOPOLOGY AND ITS APPLICATIONS   123 ( 3 ) 479 - 505  2002.09

     View Summary

    Let X be a one-dimensional metric space and H be the Hawaiian earring.
    (1) Each homomorphism from pi(1) (H) to pi(1) (X) is induced from a continuous map up to the base-point-change isomorphism on pi(1) (X).
    (2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of H if and only if pi(1) (X) is isomorphic to pi(1) (H), if and only if X has a unique point at which X is not semi-locally simply connected.
    (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if pi(1) (X) and pi(1) (Y) are isomorphic. Moreover, each isomorphism from pi(1) (X) to pi(1) (Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism. (C) 2001 Elsevier Science B.V. All rights reserved.

    DOI

  • The non-commutative Specker phenomenon in the uncountable case

    K Eda, S Shelah

    JOURNAL OF ALGEBRA   252 ( 1 ) 22 - 26  2002.06

    DOI

  • The non-commutative Specker phenomenon in the uncountable case

    K Eda, S Shelah

    JOURNAL OF ALGEBRA   252 ( 1 ) 22 - 26  2002.06

    DOI

  • On (co)homology locally connected spaces

    Katsuya Eda, Umed H. Karimov, Dušan Repovš

    Topology and its Applications   120 ( 3 ) 397 - 401  2002.05

     View Summary

    We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon. © 2002 Elsevier Science B.V. All rights reserved.

    DOI

  • On (co)homology locally connected spaces

    Katsuya Eda, Umed H. Karimov, Dušan Repovš

    Topology and its Applications   120 ( 3 ) 397 - 401  2002.05

     View Summary

    We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon. © 2002 Elsevier Science B.V. All rights reserved.

    DOI

  • Free subgroups of free complete products

    G Conner, K Eda

    JOURNAL OF ALGEBRA   250 ( 2 ) 696 - 708  2002.04

    DOI

  • Free subgroups of free complete products

    G Conner, K Eda

    JOURNAL OF ALGEBRA   250 ( 2 ) 696 - 708  2002.04

    DOI

  • The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

    Katsuya Eda

    Proceedings of the American Mathematical Society   130 ( 5 ) 1515 - 1522  2002

     View Summary

    Let X be a one-dimensional space which contains a copy C of a circle and let it not be semi-locally simply connected at any point on C. Then the fundamental group of X cannot be embeddable into a free σ-product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

    DOI

  • The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

    K Eda

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   130 ( 5 ) 1515 - 1522  2002

     View Summary

    Let X be a one-dimensional space which contains a copy C of a circle and let it not be semi-locally simply connected at any point on C: Then the fundamental group of X cannot be embeddable into a free sigma-product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

    DOI

  • アーベル群への集合論の応用

    江田 勝哉

    数学   43 ( 2 ) 128 - 138  1991

    DOI CiNii

  • Spaces which contain a copy of the rationals

    EDA Katsuya, KAMO Shizuo, NOGURA Tsugunori

    Tokyo Sugaku Kaisya Zasshi   42 ( 1 ) 103 - 112  1990

    DOI CiNii

  • SLENDER MODULES, ENDO-SLENDER ABELIAN-GROUPS AND LARGE CARDINALS

    K EDA

    FUNDAMENTA MATHEMATICAE   135 ( 1 ) 5 - 24  1990  [Refereed]

  • On Boolean powers of the group Z and (ω, ω)-weak distributivity

    EDA Katsuya, HIBINO Ken-ichi

    Tokyo Sugaku Kaisya Zasshi   36 ( 4 ) 619 - 628  1984

    DOI CiNii

  • Algebraic Topology of Peano Continua

    K.Eda

    Topology Appl.   153   317 - 328

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Research Projects

  • Embedding problems related to asymptotic dimension in coarse geometry

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2012.04
    -
    2016.03
     

    Koyama Akira, EDA Katsuya, CHINEN Naotsugu, HOSAKA Tetsuya, YAGASAKI Tatsuhiko

     View Summary

    We investigated large scale geometry, mainly coarse geometry for general proper metric spaces with coarse geometry structure and embedding problems of CAT(0)-groups into the Hilbert space. Then we discovered a good relation between coarse geometry and small scale geometry which is an ordinal geometric topology. I obtained many interesting results with the collaborators, Jozef Kransinkiewicz and Stanislaw Spiez (the Institute of Mathematic, Polish Academy of Science). Our papers have excellent evaluation and we were invited important international conferences of geometric topology. For example, Dubrovnik VIII-Geometric Group Theory and Dynamical Systems, Dubrovnik, Croatia. It is one of traditional and important topology conferences.
    Due to our proposal we organized an international meeting of coarse geometry at Wasoeda University. The future speakers are Jerzy Dydak (University of Tennessee, USA)and Stanislaw Spiez. It was a good occasion to exchange hot results and opinions.

  • Cohomological dimension theory in coarse geometry

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2009
    -
    2011
     

    KOYAMA Akira, HOSAKA Tetsuya, CHINEN Naotsugu, EDA Katsuya, YAGASAKI Tatsuhiko

     View Summary

    As we started to investigate coarse geometry from the viewpoint of asymptotic dimension theory, we faced an important problem related to embedding problems. Namely, we investigated the problem what kind of n-dimensional compact metric spaces can be embedded into a product of n one-dimensional compact metric spaces. First, we tried to determine a class of n-dimensional topological and generalized manifolds which can be embedded into a product of n one-dimensional compact metric spaces by using geometric structures and the rank of 1-dimensional cohomology groups. Next, we applied this method to arbitrary n-dimensional compact metric spaces to determine embeddability and succeeded to give a criterion by using the triviality of n-dimensional Cech cohomology groups and the rank of 1-dimensional Cech cohomology groups.

  • Ininfitely generated objects(fundamental groups of wild spaces)

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2008
    -
    2011
     

    EDA Katsuya

     View Summary

    The central theme is the non-commutative Specker phenomenon, which is generic for uncountable groups. We studied it related to group theory and algebraic topology.
    In particular we proved the following. The fundamental groups of Peano continua determine the homotopy types and each homomorphism between thosegroupsis essentially induced from a continuous map. In general the fundamental group of an arbirary Peano continuum cannot be decomposed into a non-trivial free product at wild parts. There exists a 2-dimensional, cell-like, simply-connected, non-contractible Peano continuum.

  • Infinitely generated objects

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2004
    -
    2007
     

    EDA Katsuya

     View Summary

    We have gotten results on the following five items. That is, we gotten considerably sufficient results for our original plans except for continuous words.(1) The fundamental groups of wild spaces ; (2) New constructions of wild spaces ; (3) A classification of finite-sheeted convering maps over 2-dimensional compact abelian groups ; (4) Grope groups ; (6) Infinitary words.
    (1): Suppose that a Peano continuum X is wild everywhere, i.e. not locally semi-simply connected at every point and pi_1(X) is a subgroup of the free product G^*H. Then, pi_1(X) is a conjugate of a subgroup of G or H. This implies that the fundamental groups of wild Peano continua cannot only be taken part into free products, but have a completely opposite property. This result was announced at the occasion of Borsuk conference and was submitted in December of 2005. This result is applied to a study of the fundamental groups obtained by attaching copies of the Hawaiian aearring to manifolds.
    (2) : In a collabolation with D. Repovs and U. Karimov we introduced a new construction of a space SC(X), which is obtained by attaching a cone C(X) to the square along the Topologists' sine curve. For a connected space SC(X) is simply-connected, and pi _2(4) is trivial if and only if pi_1(X) is trivial.

  • MATHEMATICAL LOGIC AND ITS APPLICATION TO COMPUTATIONAL COMPLEXITY

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2000
    -
    2002
     

    YASUMOTO Masahiro, EDA Katsuya, TSUKIJI Tatsuie, MATSUBARA Yo, OZAWA Masanao

     View Summary

    Let $N$ be a nonstandard model of $S_2$. A subset $\a$ of $N$ is called an oracle if $ (N,\a) $ satisfies $S_2 (\a) $. In this reseach we are concerned with bounded oracles $\a$ of $N$ satisfying $P^\a=NP^\a$. We proved that the existence of such oracles implies many interesting results about separations of axioms in bounded arithmetic.
    Let $n\in N$ and $M=PTC (n, \a) $ I.e.$M$ be the polynomial time closure of $\ {n\} $ with the oracle $\a$. Then it is known that $M$ is a model of $T_2^0$. Assume that there exists a bounded oracle $\a$ such that $N$ satisfies $P^\a=NP^\a$. Then $M$ satisfies Axioms $S_2$ and we proved that $M$ has no endextension satisfying $R_2^1$. This implies that $U_2^1$ is not a conservative extension of $S_2 (\a) $. In paticular, if there exists a model $N$ of $S_2$ such that $P=NP$ holds in $N$, then there is a first order sentence which is provable in $U^1_2$ but not in $S_2$. It is believed that $P\not=NP$ but there may exist a nonstandard model $N$ of $S_2$ satisfying $P=NP$.

  • CARDINAL INVARIANTS OF THE CONTINUUM AND THEIR APPLICATIONS

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2000
    -
    2001
     

    BRENDLE Jorg, EDA Katsuya, TAKAHASHI Joji, KAKUDA Yuzuru, FUCHINO Sakae, KAMO Shizuo

     View Summary

    Research in this project was devoted to the interplay between cardinal invariants of the continuum and iterated forcing theory, as well as to applications of both areas to algebra. We briefly sketch the main topics and results.
    1. Iteration along a template. Using an axiomatic approach to Shelah's recent technique of iteration along a template which was originally developed to show the consistency of σ < a, we obtained several new results. For example, we proved it is consistent that a_g>max{a,non(M)} where a_g denotes the size of the smallest maximal cofinitary group. We also showed that the almost disjointness number a can consistently be a singular cardinal of countable cofinality.
    2. Shattered iteration. Using a sophisticated new iteration technique built from a system of compete Boolean algebras which add both Cohen and random reals, we proved the simultaneous consistency of cov(M)=non(N)=N_2 and cov(N)=non(M)=c=N_3.
    3. Perfect set axioms. Say that PSP(k,Γ) holds if every set in the pointclass Γ of size at least k has a perfect subset. Let G_N_1 be the class of sets which are intersections of (at most) N_1many open sets. We showed that after iteratively adding N_2 Sacks reals over a model for CH, P5P(N_2, G_N_1) holds.
    4. Cardinal invariants related to evasion and prediction. We proved in ZFC that b【less than or equal】b_2 where b is the unbounding number and b_2 is the constant prediction number, thus answering a question of Kamo. In joint work with Shelah, we obtained several related results, for example that the k-constant prediction numbers for different k can consistently be different.
    5. The cofinality of Sym(ω). The cofinality c(Sym(ω)) of the symmetric group Sym(ω) is the least cardinal k such that Sym(ω) can be written as the union of a strictly increasing chain of proper subgroups of length k. In joint work with Losada, we proved that g【less than or equal】c(Sym(ω)) holds in ZFC, thus partially answering a question addressed by Thomas.

  • HIGHER INFINITY AXIOMS AND RELATED PROPOSITIONS OF VARIOUS FIELD OF MATHEMATICS

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1997
    -
    1998
     

    KAKUDA Yuzuru, MATSUBARA Yo, MIYAMOTO Tadatoshi, EDA Katsuya, BRENDLE Joerg, ABE Yoshihiro

     View Summary

    For properties of ideals on sets, the core of the research project, we obtained the following results by Matsubara, Abe and their cooperators ;
    1. They investigated the relation between ideals with higher infinity properties and infinite conbinatorics like t square principle, and obtained several results concerning it,
    2. They got the fact that the non-stationary ideals on Pκ(λ) cannot be precipitous under the certain assumption on the cardinality arithmetics.
    3. Abe and Shioya succeeded to characterize the fixed point of elementary embeddings defined by regular ultra filter, and showed that their result cannot be extended to the general case of uniform ultrafilters by using the forcing method.
    For forcing method, Miyamoto showed that the weak part of PFA is equiconsistent to the existence of some large cardinal.
    For properties on subsets of the reals, Blendle showed that a set of Cohen reals is either meager or empty Fuchino introduced the axiom concerning the coloring reals by using ordinals, and showed that the axiom is a generalization of the axiom introduced by Juhasz, Szentnikosse, Soukup. He also shoed the axiom holds in various models of set theory.
    For the application of axiomatic set theory to other mathematics,
    1.Eda showed that the fundamental group of the topological space subtracted lines and planes from the 3 dimensional Eucridian space is isomorphic to the subgroup of the fundamental group of Hawaiian earring.
    2. Kakuda showed that the propositional infinite conjunction and disjunction are enough for introducing the quantifier of the "existence of fictions", and succeeded to formulate the system of infinitesimals.
    4. Kakuda started to develop the mathematical formulation of the structural systems with accumulated hierarchies. This was inspired by the channel theory of Barwise, and it might be applicable not only to information theory, but also other fields, for example, the design theory of engineering.

  • Mathematical Logic

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1986
    -
    1988
     

    UESU Tadahiro, SHIRAI Kohio, UMEZAWA Toshio, EDA Katsuya, MOTOHASHI Nobuyoshi, NISHIMURA Toshio

     View Summary

    We organized five working groups. the groups exchanged informations each other, and opened meetings on several occasions. In those meetings, researchers discussed and announced results of their researches.
    In the followings, we list up the main results of the groups. Group I(Proof theory): The relations between subsystems of arithmetic with weak inductive definitions and ordinals were explained. Several results concerned with refrection pronciples and Paris-Harrington principles were obtained. Generalized built-up systems of fundamental sequences were proposed.
    Group II(Set theory and model theory): A set theory with modality and a set theory with intensionality were proposed. Assuming existence of several large cardinals, interesting abelian groups were constructed. Several properties of -stable rings were explained.
    Group III(Constructive mathematics): A theory of Fuzzy computability was made up. A consistency proof of Beeson's system PRS was given. A two-storied theory of transfinite mechanisms was proposed.
    Group IV(Theory of logical structures): It was shown that modal operators are interpretable by quantifiers in intermediate predlcate logics. The relation between the syntax and the semantics of logics without a part of structural rules, -calculus, and categorical grammars were uniformly discussed.
    Group V(Theory of non-standard universes): In a nonfinitary logic, an interpretation of infinitesimal caculus was obtained. Several properties of iterated polynomials in nonstandard analysis were explained. For Hilbert irreducibility theorem, a condition for that Z-If is finite was given, and it was shown that the bound is given by a polynomial. A nonstandard set theory, in which there are models of the five nonstandard set theories NST, IST, NSTE, NS2 and *NST, respectively, was proposed.

  • 無限生成の対象の研究(1ー2次元の野性的空間と基本群)

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • 無限生成の対象の研究(野性的空間の群論的研究)

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • 幾何学と情報数学

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • Co-operative Research of Theory of Models and related fields

  • 代数及び位相幾何における無限生成の対象の研究(数理論理学及び集合論の応用)

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • 情報伝達における数学の役割について

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • 曲面の写像類群と3次元多様体の研究

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • プログラム自動生成システム

    文部科学省 

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Misc

 

Overseas Activities

  • 無限生成の対象の研究(野性的空間の代数的位相幾何)

    2013.04
    -
    2014.03

    アメリカ   ブリガムヤング大学

    オーストリア   ウィーン工科大学

Internal Special Research Projects

  • フラクタルの基本群(非可算非可換群)

    2001  

     View Summary

    1.フラクタルの多くはペアノ空間(局所連結、連結なコンパクト距離空間)である。成果発表予定の1番目の論文の内容はペアノ空間の基本群、ホモロジー群に関する結果である。定理1:局所的 n-連結なペアノ空間 X とスレンダ-群 S、特異ホモロジ-群 H_{n+1}(X) に対して、Hom (H_{n+1}(X), S) は有限被覆のなす多面体のホモロジー群の帰納極限と同型となる。とくに、ペアノ空間 X の特異1次コホモロジー群はチェックコホモロジー群と同型である。また定理1の非可換化として、ペアノ空間 X と非可換スレンダ-群 S、基本群に対して同様の結果が成立する。定理2:ペアノ空間 X の基本群から無限個の群の自由積への準同型像は有限個の群の自由積に含まれる。準同型写像が単射であるときはそれより強い次の定理が成立する。定理3:ペアノ空間 X の基本群が2つの群の自由積 G*H の部分群と同型ならば G の有限生成部分群 G' が存在して G'*H の部分群となるか、H の有限生成部分群 H' が存在して G*H' の部分群となる。2.一般に群 G に対して、空間 X_G を定義し、次が成立することを示した。定理4:空間が1次元局所連結、連結な距離空間またはそれらの直積であるとき、G をその空間の基本群とすれば、X_G はもとの空間と同相である。